hello all
 
happy new year and hope you r having a good holiday.
 
 
i would like to calculate the expectation of a particular random variable and would like to approximate it using a number of the functions contained in R. decided to do some experimentation on a trivial example.
 
example
========
 
suppose x(i)~N(0,s2) where s2 = the variance
the prior for s2 = p(s2)~IG(a,b)
so the posterior is IG(.5*n+a, b+.5*sum(x^2) )
and the posterior expectation of s2 = (b+sum(x^2))/(.5*n+a-1)
 
I want to calculate the value of this expectation using integrals. 

ie
 
let L(X|s2) = the likelihood function
so E(s2|X) = (integral(s2*L(X|s2)*p(s2)))/(integral(L(X|s2)*p(s2))) . the bounds on both of the integrals is (0,Inf)
=========================================================================
 
 
three methods are used to calculate E(s2|X) for different sample sizes:
 
Method 1
=========
(integral(s2*L(X|s2)*p(s2)))/(integral(L(X|s2)*p(s2))) where the integrals are calculated using the integrate function
 
Method 2
=========
transform s2 to tau (from (0,Inf) to (-1,1))
f(s2) = tau = 4*arctan(s2)/pi-1
 
so 
 
E(s2|X) = (integral(jac*finv(s2)*L(X|finv(s2))*p(finv(s2))))/(integral(L(X|finv(s2))*p(finv(s2)))) .  the bounds on both integrals is (-1,1)
 
jac = the jacobian and finv is the inverse function of f(s2)
 
once again the integrals are calculated using the intergrate function
 
Method 3
=========
legendre gaussian quadriture is used to calculate each of the integrals using Method 2. 
 
 
Some results and comments are below.
method 3 seems to give the correct solution for n<=343
differences between the different methods can be detected from as early as n=3 for method 1
 
IS THERE A WAY TO MODIFY THE PROBLEM SO THAT THE EXPECTATION CAN BE CALCULATED FOR LARGE n without using MC or MCMC methods?
 
P(1)
$correct
[1] 2.201234
 
$App.2
[1] 2.201234
 
$App.trans
[1] 2.201234
$App.gq
[1] 2.201234
#=========================================
 
P(100)
$correct
[1] 18.85535
 
$App.2
[1] 15.65814
 
$App.trans
[1] 18.85028
$App.gq
[1] 18.85535
#=========================================
 

P(300)
$correct
[1] 22.59924
 
$App.2
[1] NaN
 
$App.trans
[1] 18.85405
$App.gq
[1] 22.59924
#=========================================
 
P(500)
Error in integrate(num., 0, Inf) : non-finite function value
 
 
 
The code is included below:
 

P=function(n=1,howmany=500)
{
  #you do need to have stamod
  #howmany is used to set the number of points used for the gaussian quadriture

  set.seed(1)
  sigma=5
  x=rnorm(n)*sigma
  a=5
  b=5
  
  #the correct expectation
  #===========================================================
  correct=(b+sum(x^2)*.5)/(.5*n+a-1)
  
 
  #Method 1
  #===========================================================
  num.=function(s2)
  {
   L=((2*pi*s2)^(-n*.5))*exp(-.5*sum(x^2)/s2)
   P=(s2^(-(a+1)))*exp(-b/s2)*(b^a)/gamma(a)
   return(s2*L*P)
  }
  den.=function(s2)
  {
   L=((2*pi*s2)^(-n*.5))*exp(-.5*sum(x^2)/s2)
   P=(s2^(-(a+1)))*exp(-b/s2)*(b^a)/gamma(a)
   return(L*P)
  }
  App.2=integrate(num.,0,Inf)$value/integrate(den.,0,Inf)$value
  
 
  #Method 2
  #===========================================================
  num.t=function(s2.)
  {
    s2=tan(.25*pi*(1+s2.))
    jac=.25*pi*(((tan(.25*pi*(1+s2.)))^2)+1)
    num.tv=jac*exp(-(a+.5*n)*log(s2)-(b+sum(x^2)*.5)/s2)
    return(num.tv)
  }
  den.t=function(s2.)
  {
    s2=tan(.25*pi*(1+s2.))
    jac=.25*pi*(((tan(.25*pi*(1+s2.)))^2)+1)
    den.tv=jac*exp(-(a+.5*n+1)*log(s2)-(b+sum(x^2)*.5)/s2)
    return(den.tv)
  }
  App.trans=integrate(num.t,-1,1)$value/ integrate(den.t,-1,1)$value
 

  #Method 3
  #===========================================================
  library(statmod)
  ad.=gauss.quad(n=howmany,kind="legendre",alpha=-1,beta=-1)
  n.=ad.$nodes
  w.=ad.$weights                                        
  App.gq=sum(w.*num.t(n.))/sum(w.*den.t(n.))
  
  list(correct=correct,App.2=App.2,App.trans=App.trans, App.gq=App.gq)
}
 
 
Allan Clark
========
Lecturer in Statistical Sciences Department
University of Cape Town
7701 Rondebosch
South Africa
TEL (Office): +27-21-650-3228
FAX: +27-21-650-4773
http://web.uct.ac.za/depts/stats/aclark.htm 
 


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